Problem 21 · 2013 Math Kangaroo
Stretch
Fractions, Decimals & Percents
fraction-to-decimalplace-value
How many decimal places are necessary to write the number \(\frac{1}{1024000}\) as a decimal?
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Answer: C — 13
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Hint 1 of 2
A fraction terminates after as many places as the larger of the powers of 2 and 5 in the denominator.
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Hint 2 of 2
Factor 1024000 into 2's and 5's.
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Approach: count powers of 2 and 5 in the denominator
- 1024000 = 1024 × 1000 = 2¹⁰ × (2³ · 5³) = 2¹³ · 5³.
- A terminating decimal needs the denominator to become 10^k with k = max(13, 3) = 13.
- So 13 decimal places are required.
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